Experimental design postulated model

To emphasize this point, once a model structure is postulated, the compartmentai matrix is known, since it depends only upon the transfers and losses. The input, the F q, comes from the experimental input and thus is determined by the investigator. In addition, the units of the differential equation (i.e., the units of the Xj) are determined by the units of the input. The point is that if the parameters of the model can be estimated from the data from a particular experimental design li.e., if the model is a priori identifiable see... 

There are no classic experimental designs that exist for such circumstances, and a purely empirical approach is required 1) to postulate a mathematical model that is expected to describe the response and 2) to then select from among the many possible experiments a design that will determine the model coefficients with maximum efficiency. 

Another solution, by far the more general one, is to carry out the experiments in a random order. If there are the same number of experiments in the design as there are parameters in the postulated model, the effect of time is to perturb the different estimations in a random fashion. If the number of experiments exceeds the number of parameters to be estimated then the experimental error estimated by multi-linear regression (see chapter 4) includes the effect of time and is therefore overestimated with respect to its true value. 

I) poiiiLs. ill the cxperiiiteiital doftiain of interest, situated far fron) tlie points of the experimental design. This will enable us to test the validity of the model postulated. 

Very often one single experimental design does not lead to the solution of the problem. In those cases the information obtained at point 5 is used to reformulate the problem (removal of the non-significant variables, redefinition of the experimental domain, modification of the postulated model), after which one... 

Figure 10 shows the leverage plot of this experimental design. The leverage can be computed at every point of the experimental domain (it depends on the experimental matrix and on the postulated model), and its value. 

One goal of our experimental program with the bench-scale unit was to develop the necessary correlations for use in the ultimate design of large commercial plants. Because of the complexity inherent in the three-phase gas-liquid-solid reaction systems, many models can be postulated. In order to provide a background for the final selection of the reaction model, we shall first review briefly the three-phase system. 

To develop the rate equations suitable for process modeling and reactor design, experimental data have been analyzed on the basis of the postulated reaction mechanism [2] given in Table 1. Here the formation of polymer is excluded because it is not detected under our experimental conditions. All of the reactions are equilibrium-limited and the net rates for the formation of each component with some assumptions [3] are given as follows ... 

The next step in the development of the Bohr model was his assertion that the angular momentum of the electron is quantized. This was an ad hoc assumption designed to produce stable orbits for the electron it had no basis in either classical theory or experimental evidence. The linear momentum of an electron is the product of its mass and its velocity, mgV. The angular momentum, L, is a different quantity that describes rotational motion about an axis. An introduction to angular momentum is provided in Appendix B. For the circular paths of the Bohr model, the angular momentum of the electron is the product of its mass, its velocity, and the radius of the orbit (L = meVr). Bohr postulated that the angular momentum is quantized in integral multiples of / /2tt, where h is Planck s constant ... 

And, as we have already seen a number of times, the experimenter does not know what will be the best model. He therefore has the following dilemma to set up the design he must postulate a model, but if he makes the wrong choice of model, the design loses much of its interest. 

E.xtension of the number of variables studied. It is always po.ssible to add other variables to those under study without the design losing its quality. Let us suppose that our experimental domain is dehned by 7 factors and that we can run only 3 experiments in the course of a week. If we postulate a second-degree polynomial model, the number of experiments is abtnil 60. which will require about 20-25 weeks work. It is veiy difficult to remain isolated for 6 month.s to obtain the totality of the experimental responses in order to perform the inierpreta-... 

It is advised to add test points to the simplex lattice designs. These points should help test the validity of the model postulated. We advise adding the points situated inside the experimental domain corresponding to mixtures containing all the components. 

Van Parijs et al. [1986] applied sequential discrimination in their experimental study of benzothiophene hydrogenolysis. The final rate equations were already given in Example 2.6.4.A. The efficiency of the sequential discrimination was remarkable. At 533 K, a total of seven experiments (four preliminary, three designed) were sufficient to reject 14 out of 16 rivals. Only one of the two remaining models was consistently the best at 513, 553, and 573 K also, after four, five and nine designed experiments, respectively. A total of 41 experiments sufficed to select the best among the postulated rate equations. The location of the settings of these experiments is shown in Fig. 2.7.2.2.A-1 of Example 2.7.2.2.A. 

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